Operator system

Given a unital C*-algebra [math]\displaystyle{ \mathcal{A} }[/math], a *-closed subspace S containing 1 is called an operator system. One can associate to each subspace [math]\displaystyle{ \mathcal{M} \subseteq \mathcal{A} }[/math] of a unital C*-algebra an operator system via [math]\displaystyle{ S:= \mathcal{M}+\mathcal{M}^* +\mathbb{C} 1 }[/math].

The appropriate morphisms between operator systems are completely positive maps.

By a theorem of Choi and Effros, operator systems can be characterized as *-vector spaces equipped with an Archimedean matrix order.

See also

• Operator space